Polar Codes for Arbitrary Classical-Quantum Channels and Arbitrary cq-MACs

We prove polarization theorems for arbitrary classical-quantum (cq) channels. The input alphabet is endowed with an arbitrary Abelian group operation, and an Arıkan-style transformation is applied using this operation. It is shown that as the number of polarization steps becomes large, the synthetic cq-channels polarize to deterministic homomorphism channels that project their input to a quotient group of the input alphabet. This result is used to construct polar codes for arbitrary cq-channels and arbitrary cq multiple access channels. The encoder can be implemented in
O
(
N
log
N
) operations, where
N
is the blocklength of the code. A quantum successive cancellation decoder for the constructed codes is proposed. It is shown that the probability of error of this decoder decays faster than 2(-N)
^β
for any β <; (1/2).